10.6.2 Tensor Inverse Basis Target Frame

Tensor Inverse Basis Target Frame is the designation given to the original basis, that is, the basis that existed before the forward change of basis was applied, when that basis is treated as the destination reached by the inverse basis change rule. It names the frame at which the inverse transformation arrives, complementing the inverse basis source frame from which the same transformation departs, and its purpose is to remove any ambiguity about which of the two bases involved in a change of basis is being recovered by a given inverse computation.

Role Within the Inverse Rule

Identifying the Destination of the Reverse Transformation

Every application of the inverse basis change rule carries a quantity expressed in the source frame to an equivalent expression in the target frame. The target frame in this reverse direction is always the original unprimed basis, since it is precisely this basis that the inverse rule is designed to reconstruct.

Here the unprimed basis vector on the left is the inverse basis target frame, since it is the quantity being produced by the inverse rule from the primed source frame on the right.

Free Index Placement

The inverse basis target frame is marked in a formula by the free index attached to the unprimed symbol, which matches the upper index of the inverse coefficient matrix. This index placement is what allows the target frame quantity to be read directly off the left-hand side of any inverse transformation equation.

Properties of the Target Frame

Recovering the Pre-Transformation Basis

The inverse basis target frame is not a newly introduced basis; it is exactly the basis that was already in use before the original forward change was carried out. Its properties, including its own relationships to any further bases, are unaffected by serving as the target of the inverse rule.

Anchoring Point for Consistency Checks

Because the inverse basis target frame is known independently, from having been the starting basis of the original forward change, it provides a fixed reference against which the result of the inverse computation can be checked. If the components or basis vectors produced by the inverse rule do not match this known target frame, an error exists somewhere in the coefficient matrix or in the application of the transformation.

Role in Chained Basis Changes

In a sequence of successive basis changes, the inverse basis target frame at any given step is the basis immediately preceding the current one in the chain. Each step of an inverse chain therefore has its own target frame, distinct from the ultimate original basis unless the chain has only a single link.

Schematic Representation

The diagram places the target frame on the left, representing the original basis being recovered, with the arrow of the inverse basis change rule pointing from the source frame on the right into this target frame.

Distinction From the Forward Target Frame

Reversal of Roles

In the forward change of basis, the new primed basis serves as the target and the original unprimed basis serves as the source. The inverse basis target frame reverses this assignment entirely: the unprimed basis, previously the source, becomes the target of the inverse rule.

Necessity for Correct Reconstruction

Correctly identifying the inverse basis target frame ensures that the components and basis vectors produced by applying the inverse matrix are interpreted as belonging to the original basis rather than mistakenly treated as a further, unrelated basis, which would break the cancellation property linking the forward and inverse rules.